User:PAR/Work6

The presence of pertubing particles near an emitting atom will cause a broadening and possible shift of the emitted radiation.

There is two types impact and quasistatic

In each case you need the profile, represented by Cp as in C6 for Lennard-Jones potential

:$\backslash gamma\; =\; \backslash frac\{C\_p\}\{r^p\}$

Assume Maxwell-Boltzmann distribution for both cases.

From {{harv|Peach|1981|p=387}}

For impact, its always Lorentzian profile

:$P(\backslash omega)=\backslash frac\{1\}\{\backslash pi\}~\backslash frac\{w\}\{(\backslash omega-\backslash omega\_0-d)^2+w^2)\}$

:$w+id=\backslash alpha\_p\; \backslash pi\; n\; v\backslash left[\backslash frac\{\backslash beta\_p|C\_p|\}\{v\}\backslash right]^\{2/(p-1)\}$

\Gamma\left(\frac{p-3}{p-1}\right)\exp\left(\pm \frac{i\pi}{p-1}\right)

:$\backslash alpha\_p=\backslash Gamma\backslash left(\backslash frac\{2p-3\}\{p-1\}\backslash right)\backslash left(\backslash frac\{4\}\{\backslash pi\}\backslash right)^\{1/(p-1)\}$

:$v=\backslash sqrt\{\backslash frac\{8kT\}\{\backslash pi\; m\}\}$

:$\backslash beta\_p=\backslash frac\{\backslash sqrt\{\backslash pi\}\backslash ,\backslash Gamma((p-1)/2)\}\{\backslash Gamma(p/2)\}$

• Linear Stark p=2

:Broadening by linear Stark effect

:$\backslash gamma=divergent$

:$C2=???$

:Debye effects must be accounted for

• Resonance p=3

:Broadening by ???

:$\backslash gamma=divergent$

:$C3=???$

• Quadratic Stark p=4

:Broadening by quadratic Stark effect

:$\backslash gamma=divergent$

:$C4=???$

• Van der Waals p=6

:Broadening by Van der Waals forces

:$\backslash gamma=divergent$

:$C6=???$

From {{harv|Peach|1981|p=408}}

For quasistatic, functional form of lineshape varies. Generally its a Levy skew alpha-stable distribution (Peach, page 408)

:$\backslash Delta\backslash omega\_0\; L(\backslash omega)=\backslash frac\{1\}\{\backslash pi\}\backslash Re\backslash left[\backslash int\_0^\backslash infty$

\exp(i\beta x-(1+i\tan\theta)x^{3/p})\,dx\right]

:$\backslash beta=\backslash Delta\backslash omega/\backslash Delta\backslash omega\_0\backslash ,$

:$\backslash Delta\backslash omega=\backslash omega-\backslash omega\_0\backslash ,$

:$\backslash theta=\backslash pm\; 3\backslash pi/2p\backslash ,$

:$\backslash Delta\backslash omega\_0=|C\_p|\backslash left(\backslash frac\{4\backslash pi\; n\}\{3\}\backslash Gamma(1-3/p)\backslash cos(\backslash theta)\backslash right)^\{p/3\}$

• Linear Stark p=2

:Broadening by linear Stark effect

:$P(\backslash nu)=\backslash frac\{1\}\{\backslash pi\backslash gamma\}\backslash int\_0^\backslash infty$

\cos\left(\frac{x(\nu-\nu_0)}{\gamma}\right)\exp(x^{-3/2})\,dx

:$\backslash gamma=|C\_2|\backslash pi\backslash left(\backslash frac\{32n^2\}\{9\}\backslash right)^\{1/3\}$

:$C2=???$

• Resonance p=3

:$P(\backslash omega)=\backslash frac\{\backslash gamma\}\{(\backslash omega-\backslash omega\_0)^2+\backslash gamma^2\}$

:$\backslash gamma=|C\_3|2\backslash pi^2n/3\backslash ,$

:$C\_3=K\backslash sqrt\{\backslash frac\{g\_u\}\{g\_l\}\}~\backslash frac\{e^2f\}\{2m\backslash omega\}$

where K is of order unity. Its just an approximation.

• Quadratic Stark p=4

:Broadening by quadratic Stark effect

:$P(\backslash nu)=???$

:$\backslash gamma=|C\_4|\backslash left(\backslash frac\{4\backslash pi\}\{3\}\backslash Gamma(1/4)\backslash cos(\backslash theta)n\backslash right)^\{4/3\}$

:$C\_4=-\backslash frac\{e^2\}\{2\backslash hbar\}(\backslash alpha\_i-\backslash alpha\_j)$ {{harv|Peach1981|Peach|1981|loc=Eq 4.95}}

where $\backslash alpha\_i$ and $\backslash alpha\_j$ are the static dipole polarizabilities of the i and j energy levels.

:$\backslash theta=\backslash pm\; \backslash frac\{3\backslash pi\}\{8\}$

• Van der Waals p=6

:Broadening by Van der Waals forces gives a Van der Waals profile. C6 is the wing term in the Lennard-Jones potential.

:$P(\backslash omega)=\backslash sqrt\{\backslash frac\{\backslash gamma\}\{2\backslash pi\}\}~$

\frac{\exp\left(-\frac{\gamma}{2|\nu-\nu_0|}\right)}{(\nu-\nu_0)^{3/2}}

for

:$(\backslash nu-\backslash nu\_0)C\_6\backslash ge\; 0$

0 otherwise.

:$\backslash gamma=|C\_6|\backslash frac\{8\backslash pi^3n^2\}\{9\}\backslash ,$

:$\backslash Delta\; \backslash omega\_0=\backslash frac\{\backslash pi^4\; n^2\}\{9\}|C\_6|\backslash ,$ {{harv|Peach|1981|loc=Eq 4.101}}

:$C\_6=-K\backslash frac\{\backslash mu\_1^2\}\{\backslash hbar\}(\backslash alpha\_i-\backslash alpha\_j)$ {{harv|Peach|1981|loc=Eq 4.100}}

where K is of order 1.